On Sep 28, 9:04 am, Bill Hart <[EMAIL PROTECTED]> wrote: > It would truly be interesting to do a profile of a complex interval > approach against a purely algebraic approach. It sounds like a > question begging to be answered.
Yes. (Along with a purely interval approach, with the root separation bounds.) > I notice that in the example William gave, defining complex intervals > as AA[x]/(x^2+1) he was able to find minimum polynomials. I'm curious > as to how SAGE does that and whether the result is always reliable. William's example actually finds the minimal polynomials for the algebraic reals which are the real and imaginary components of the given complex number. The canonical representation of an algebraic real (which is computed lazily, if interval arithmetic does not suffice) is as a standard SAGE NumberFieldElement, along with an embedding from that NumberField into R; the minpoly() method on an algebraic real just calls the minpoly() method on the underlying NumberFieldElement. So it is always reliable, but may be slow (because it uses algebraic methods). There's another way to do this, which does not compute the NumberFieldElement and is not reliable: sage: (z1^10)[1].minpoly() x^4 - 17754903/2*x^2 + 75340716089409/16 sage: RealIntervalField(300)((z1^10)[1]).algdep(4).monic() x^4 - 17754903/2*x^2 + 75340716089409/16 The first line is from William's example. The second line computes a (guaranteed accurate) 300 bit floating-point approximation of the number in question, and uses Pari to find a polynomial with that floating-point number as a root. But this is unreliable, since you need to guess both the number of bits to use (for this example, 300 bits works, but 200 is too few) and the degree of the desired polynomial (the "4" which is a parameter to algdep). Carl --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
